Problem: Simplify the following expression and state the conditions under which the simplification is valid. You can assume that $y \neq 0$. $a = \dfrac{-2y - 20}{y + 5} \times \dfrac{y^2 - 25}{-4y^2 - 76y - 360} $
First factor out any common factors. $a = \dfrac{-2(y + 10)}{y + 5} \times \dfrac{y^2 - 25}{-4(y^2 + 19y + 90)} $ Then factor the quadratic expressions. $a = \dfrac {-2(y + 10)} {y + 5} \times \dfrac {(y + 5)(y - 5)} {-4(y + 10)(y + 9)} $ Then multiply the two numerators and multiply the two denominators. $a = \dfrac {-2(y + 10) \times (y + 5)(y - 5) } {(y + 5) \times -4(y + 10)(y + 9) } $ $a = \dfrac {-2(y + 5)(y - 5)(y + 10)} {-4(y + 10)(y + 9)(y + 5)} $ Notice that $(y + 10)$ and $(y + 5)$ appear in both the numerator and denominator so we can cancel them. $a = \dfrac {-2(y + 5)(y - 5)\cancel{(y + 10)}} {-4\cancel{(y + 10)}(y + 9)(y + 5)} $ We are dividing by $y + 10$ , so $y + 10 \neq 0$ Therefore, $y \neq -10$ $a = \dfrac {-2\cancel{(y + 5)}(y - 5)\cancel{(y + 10)}} {-4\cancel{(y + 10)}(y + 9)\cancel{(y + 5)}} $ We are dividing by $y + 5$ , so $y + 5 \neq 0$ Therefore, $y \neq -5$ $a = \dfrac {-2(y - 5)} {-4(y + 9)} $ $ a = \dfrac{y - 5}{2(y + 9)}; y \neq -10; y \neq -5 $